[mannykannot]

"So, in Borel’s view, most reals, with probability one, are mathematical fantasies, because there is no way to specify them uniquely."

If that were so, might it not offer a way to an "explanation" for the Banach-Tarski paradox that even the likes of I could imagine I understood? - that duplicate volume you constructed is made from the same reals, specified differently!

Not that that would be an argument for the proposition quoted.

I was going to say that I am not a mathematician, but that would be redundant.

[eli_gottlieb]

Banach-Tarski isn't a paradox. It's just what happens when you build geometric "shapes" out of nonmeasurable sets. In actual, real-world geometry, we almost always assume the set of points inside the shape/construction we care about is measurable, and that any geometric operations we perform must preserve measure.

[mannykannot]

Thanks - I have seen it mentioned in many places (often with "paradox" appended, e.g. [1]), but never with such a succinct commentary as you have given here.

[1] http://mathworld.wolfram.com/Banach-TarskiParadox.html